# circular helix

The space curve traced out by the parameterization

 $\boldsymbol{\gamma}(t)=\left[\begin{array}[]{c}a\cos(t)\\ a\sin(t)\\ bt\end{array}\right],\quad t\in\mathbb{R},\;a,b\in\mathbb{R}$

is called a circular helix (plur. helices).

Its Frenet frame is:

 $\displaystyle\mathbf{T}$ $\displaystyle=\frac{1}{\sqrt{a^{2}+b^{2}}}\begin{bmatrix}-a\sin t\\ \hphantom{-}a\cos t\\ b\end{bmatrix}\,,$ $\displaystyle\mathbf{N}$ $\displaystyle=\begin{bmatrix}-\cos t\\ -\sin t\\ 0\end{bmatrix}\,,$ $\displaystyle\mathbf{B}$ $\displaystyle=\frac{1}{\sqrt{a^{2}+b^{2}}}\begin{bmatrix}\hphantom{-}b\sin t\\ -b\cos t\\ a\end{bmatrix}\,.$

Its curvature and torsion are the following constants:

 $\displaystyle\kappa=\frac{a}{a^{2}+b^{2}}\,,\quad\tau=\frac{b}{a^{2}+b^{2}}\,.$

A circular helix can be conceived of as a space curve with constant, non-zero curvature, and constant, non-zero torsion. Indeed, one can show that if a space curve satisfies the above constraints, then there exists a system of Cartesian coordinates in which the curve has a parameterization of the form shown above.

An important property of the circular helix is that for any point of it, the angle $\varphi$ between its tangent and the helix axis is constant. Indeed, if we consider the position vector of that arbitrary point, we have (where $\mathbf{k}$ is the unit vector parallel to helix axis)

 $\displaystyle\frac{d\boldsymbol{\gamma}}{dt}\cdot\mathbf{k}=\begin{bmatrix}-a% \sin t\\ \hphantom{-}a\cos t\\ b\end{bmatrix}\begin{bmatrix}0\;0\;1\end{bmatrix}=b\equiv\bigg{\lVert}\frac{d% \boldsymbol{\gamma}}{dt}\bigg{\rVert}\cos\varphi=\sqrt{a^{2}+b^{2}}\cos\varphi.$

Therefore,

 $\displaystyle\cos\varphi=\frac{b}{\sqrt{a^{2}+b^{2}}}\text{constant},$

as was to be shown.

There is also another parameter, the so-called pitch of the helix $P$ which is the separation between two consecutive turns. (It is mostly used in the manufacture of screws.) Thus,

 $\displaystyle P=\gamma_{3}(t+2\pi)-\gamma_{3}(t)=b(t+2\pi)-bt=2\pi b\,,$

and $P$ is also a constant.

Title circular helix CircularHelix 2013-03-22 13:23:25 2013-03-22 13:23:25 rspuzio (6075) rspuzio (6075) 13 rspuzio (6075) Definition msc 53A04 SpaceCurve RightHandedSystemOfVectors circular helices